383] 
PROBLEMS AND SOLUTIONS. 
597 
and hence writing (a, /3, y) = (0, —1, 1) we have the asymptote x+2y + 2z = Q: to find 
where this meets the curve, we have 6 i -f 4# 3 + 4# 2 —1 = 0, that is (9 4- l)' 2 (9' 2 4- 29 — 1) = 0, 
or at the points of intersection 9 2 4- 29 — 1 = 0, that is 9 = — 1 4 *J2, or there are two 
real points of intersection. 
Again writing (a, /3, <y) = (—l—k, k, 1) we find the asymptote k 2 x—2y + (k + l)z = 0: 
to find where this meets the curve, we have №(9* — 1) - 4<k9 3 + (2k 4- 2) & 2 = 0, that is 
lc 2 9 4 — 4>k9 3 + (2k 4- 2) & 2 — k? = (9 — Jc)* [k 2 9' 2 — 2 (k 2 + k +1) 9 — 1} = 0; or for the points of 
intersection l?& 2 — 2 (k 2 + k + 1) 9 — 1 = 0, an equation in 9 with two real roots, hence 
the points of intersection are real. 
It is now easy to lay down the curve; viz., if, to fix the ideas, the fundamental 
triangle is taken to be equilateral, and the coordinates x, y, z are considered to be 
positive for points within the triangle, then the curve is as shown in the annexed 
figure 1. 
It may be remarked that the curve is met by every real line in two real points 
at least, and consequently that it is not the projection of any finite curve whatever. 
By g, modification of the constants of the equation, we might obtain curves which are 
finite, such as the curve in figure 2 ; or curves with two or four infinite branches, 
which are the projections of such a finite curve. 
[Vol. iv. pp. 32—37.] 
1744. (Proposed by W. S. Burnside, B.A.)—It is required to find (x u y l} z-9), 
functions of {x, y, z), such that we may have identically 
xi 2 + yi 2, + zi 2 _ x^ + y 3 + z 2 
Xiy 1 z 1 “ xyz 
Solution by Professor Cayley. 
The Solution is in fact given in my “ Memoir on Curves of the Third Order,” 
Philosophical Transactions, vol. cxlvii. (1857), pp. 415—446, [146]. 
_J_ ^/3 _|_ ^3 
Write = — 61] then, taking (X, Y, Z) as current coordinates, (x, y, z) are, 
xyz 
it is clear, the coordinates of a point on the cubic curve X 3 + Y' s + Z 3 + QIXYZ = 0 •,